3.2.2. Temperature and Velocity Profiles

Figure 7 shows the temperature distribution along the gap between the inner and outer hexagonal cylinders at three di fferent angular directions, θ = 0◦ (along the line passing through the vertical median), θ = 30◦ (along the line passing through the right uppermost corners of the two hexagons), and at θ = 90◦ (along the line passing through the horizontal median) when *AR* = 0.2. *Ri* and *Ro* in the figure denote the radius of circles that pass through the edges of inner and outer hexagonal cylinders, respectively. The temperature distribution profiles are plotted for two Rayleigh numbers: Ra = 10<sup>3</sup> and Ra = 106. The symbols in the figure indicate the corresponding results obtained from the Fluent, which are in good agreemen<sup>t</sup> with the present simulation results. When Ra = 103, the temperature profiles, along all three θ directions, show a quasi-linear pattern (in other words, the temperature gradients along the gap are almost constant) with the gap as the conduction is the primary mode of the heat transfer process in the annulus. The temperature profiles along θ = 30◦ and θ = 90◦ are almost the same as the isothermal lines are almost concentric and *Ri* values at θ = 30◦ and θ = 90◦ are the same. When Ra = 106, as the strong convective fluid flow disturbs the uniform temperature distribution over the inner hexagon, high-temperature gradients are observed near the inner and outer hexagonal cylinder edges. As mentioned earlier, a thermal plume is formed in the direction of θ = 0◦ for *AR* = 0.2, the temperature gradients closer to the outer cylinder wall are very steep for the temperature profile along θ = 0◦ as the thermal boundary layer thickness is very thin due to continuous impingement thermal plume against the top flat edge of the outer hexagonal cylinder. The slope of the temperature profile near the outer cylinder wall is steeper for θ = 30◦ compared to that for θ = 90◦ as the convective fluid flow intensity is weaker at θ = 90◦.

**Figure 7.** Temperature distribution along the gap between the inner and outer hexagonal cylinders at di fferent angular directions, θ = 0◦, θ = 30◦, and at θ = 90◦ for *AR* = 0.2. The symbols represent corresponding data from Fluent software.

The curves for the temperature distributions along the gap when *AR* = 0.6 are provided in Figure 8 for Rayleigh numbers, Ra = 10<sup>3</sup> and Ra = 106. The temperature data obtained from Fluent software is also provided (the symbols in the figure) for comparison purposes. The agreemen<sup>t</sup> between the two results is excellent, implying the capability of the present simulation technique in the simulation of fluid flow and heat transfer in the hexagonal annuals. In this case, also the slope of the temperature profiles in each θ direction is almost constant when Ra = 103. For Ra = 10<sup>6</sup> case, the temperature profile in the direction of θ = 30◦ is in a similar trend with that of θ = 0◦ of Figure 7 data (for the case of *AR* = 0.2) as the formation of thermal plume for *AR* = 0.6 is along the direction of θ = 30◦. The slope of the temperature profile near the inner cylinder wall is very steep for θ = 0◦ as thermal boundary layer thickness over the flat top edge of the inner cylinder is very small due to the formation of the thermal plume in the reverse direction.

**Figure 8.** Temperature distribution along the gap between the inner and outer hexagonal cylinders at angular directions, θ = 0◦, θ = 30◦, and at θ = 90◦ for *AR* = 0.6. The symbols represent corresponding data from Fluent software.

The heat transfer between the outer and inner cylinders is enhanced by convection mode through fluid circulation. The rotational velocity (tangential velocity), *u*θ, can be used as a good indication for the intensity of the convective fluid flow. Figure 9 shows the variations of tangential velocity, *u*θ, along the gap between the inner and outer hexagonal cylinders for Rayleigh numbers, Ra = 10<sup>3</sup> and Ra = 106, and for aspect ratio, *AR* = 0.2. The profiles are plotted for θ = 30◦ and θ = 90◦. The reference velocity, α/(*Ro* − *Ri*), has chosen to normalize *u*θ. It is noted that the magnitudes of *u*θ when Ra = 10<sup>3</sup> are very small (almost zero) compared to those when Ra = 10<sup>6</sup> because of a weak fluid flow intensity at low Ra. For the velocity profile at θ = 90◦, the location of the flow reversal point is exactly at the center of the gap (at the halfway between the corners of two hexagons). However, the flow inversion point for the profile at θ = 30◦ is located a bit away from the gap center towards the outer cylinder. The velocity gradients for the profile at θ = 90◦ are steeper near the inner cylinder than those at the outer cylinder as the convective currents in the region adjacent to the inner cylinder (where fluid gets heated) are stronger than those near the outer cylinder.

**Figure 9.** Variation of tangential velocity, *u*θ, along the gap between the inner and outer hexagonal cylinders at angular directions, θ = 30◦, and θ = 90◦ for *AR* = 0.2. The symbols represent the corresponding data from Fluent software.

The profiles for the tangential velocity distributions when *AR* = 0.6 are provided in Figure 10 for Rayleigh numbers, Ra = 10<sup>3</sup> and Ra = 106. In this case, as well the magnitudes of *u*θ when Ra = 10<sup>3</sup> are very small compared to those when Ra = 106. When *AR* = 0.6, as the thermal plume is formed along the θ = 30◦ direction, the tangential velocity in that direction is very low (fluid flow radially outwards along θ = 30◦). Therefore, the magnitude of *u*θ along θ = 30◦ very low compared to that along θ = 90◦. The locations for the flow reversal points of the two velocity profiles along θ = 30◦ and θ = 90◦ are the same and are located away from the gap center and are towards the inner cylinder. The magnitudes of *u*θ obtained for *AR* = 0.6 case are lower compared to those obtained for *AR* = 0.2 (Ra = 10<sup>6</sup> data of Figure 9) as available space for convection flow is constricted at *AR* = 0.6. The tangential velocity profiles data obtained from the present simulation technique, for cases of *AR* = 0.2 and *AR* = 0.6, and Ra = 10<sup>3</sup> and Ra = 106, are compared with those of Fluent software (the symbols in Figures 9 and 10). The present simulation results show good agreemen<sup>t</sup> with the Fluent data.

**Figure 10.** Variation of tangential velocity, *u*θ, along the gap between the inner and outer hexagonal cylinders at angles, θ = 30◦, and θ = 90◦ for *AR* = 0.6. The symbols represent the corresponding data from Fluent software.
